Relatively bounded and compact perturbations of nth order differential operators

RELATIVELY

BOUNDED

AND COMPACT PERTURBATIONS

OF

NTH ORDER

DIFFERENTIAL OPERATORS

TERRYG.ANDERSON

Department

ofMathematical Sciences

Appalachian State University

Boone, NC

28608, U.S.A.

E-mailaddress: [email protected]

(Received June I0, 1996 and in revised formAugust 26, 1996)

ABSTRACT.

Aperturbationtheoryfor nth order differential operators isdeveloped.

For

certain

classes

ofoperators

L,

necessaryand sufficient conditions are obtained for aperturbing operator

B

to be relatively bounded or relatively compact withrespecttoL. Theseperturbationconditions involve

explicit integral averagesof the coefficients ofB. The

proofs

involveinterpolation inequalities.

KEY

WORDS AND PHRASES.

Perturbationtheory, differential operators,relativelybounded, relativelycompact, integral averages, interpolation inequalities,maximal and minimaloperators,

essential spectrum, Fredholm index.

1991

AMS

SUBJECT

CLASSIFICATION CODES.

34L99, 47E05.

INTRODUCTION AND MAIN RESULTS

We develop a perturbation theory for nth order differential operators.

In

the following, the differential operatorBwillberegardedas aperturbation of a(typically)higher-order differential operatorL.

For

certainclasses of operatorsL,we obtainnecessaryand sufficient conditions for

B

to be L-bounded or

L-compact. We

employthefollowing terminologyas given inKato [5,

pp.

190,

194].

DEFINITION

A.

B

isrelatively boundedwithrespectto

L

or simply L-bounded if D(L)

_

D(B)

and

B

isbounded onD(L) withrespecttothegraphnorm

II.

II,.

ofLdefinedby

Ilyll,.

Ilyll

/

llLyll,

y e D(L), whereD(L) denotes the domain ofL.

In

otherwords,

B

isL-bounded ifD(L) D(B)

and thereexistnonnegative constantsct and

fl

such that

y D(L).

Thegreatestlowerbound

/0

ofallpositive

constants/

forwhich thisinequalityholds is called the relativebound

of

B

with respectto

L

orsimplytheL-bound

of

B.

In

general,theconstant

o

will increasewithout bound as

/

is chosencloser

to/0

(sothat the infimum

/0

neednotbeattained).

A

sequence

{y,

}

issaidtobeL-bounded if thereexists K > 0 such that

Ily.ll,

<

.

_>

.

B

iscalled relatively compactwithrespectto

L

orsimply

L-compact

if D(L) D(B)and

B

sequencewhich has a convergent

subsequence. For

example, if

L

is the identity

map,

then

L-boundedness (L-compactness) of B is equivalent to the usual operator norm boundedness

(compactness)ofB.

The functionspace settingistheweightedBanach

space

/.(I),

where _< p < .o, Wisa

positive Lebesguemeasurable function defined on an interval

I

of the real line, and

/.

(I)

denotes

theLebesguespaceof equivalence classes ofcomplex-valuedfunctionsywith domain

I

such that

IIMI

:=

[],

w

lye"

]""

< .o. If

w

1, we denote this

space

by

L"(1).

The

space

of

complex-valued functionsywith domain

I

suchthat

I11.

:= =ss

sop

[y(t)

< isdenoted by

L’(1). A

localproperty is indicatedbyuseofthe subscript"loc," and

AC

is usedtoabbreviateabsolutely

continuous. Thespaceof allcomplex-valued,ntimescontinuouslydifferentiable functions on

I

is denoted by

C"(1)"

C(1)

denotes the restriction of

C"(1)

to functions withcompact support

contained in

I;

and

C0(l

isthe

space

of allcomplex-valuedfunctions on

I

which areinfinitely

differentiableandhave compactsupportcontained inthe interior ofI. We adoptthe definitionsof maximaland minimaloperators giveninGoldberg [4,

pp.

127-128,

135].

DEFINITION B.

Let be a differential expression of the form

Wilp

ai(t)

(D

),

where

W

is a positive

Lebesgue

measurable function defined on l and each

a,

is a

complex-valuedfunction onI. Then the maximal operator

L

corresponding to has domain

D(L)

{

y

I.(I)"

y-"

AC(1),

/[y] e

/,(I)}

and action

L[y] l[y]

,__oa,(t)

(y

D(L)).

If

a,

.

Cl(1)

for 0 < < n and

a,

0onl,

then the minimal operator

L

correspondingto is definedtobe the minimal closed extension of

L

restricted tothose y D(L) whichhave compactsupportin the interiorof

L

In

the Hilbertspace settingof

L=(1),

mostof the smoothness requirements on the coefficients

a,

(0 < < n) are not needed, and thetheoryisdevelopedin Naimark[7,sect.17].

We

considerperturbations

n-!

B

W,I

_j=O

E

bj/T

(a < < to)

of the operators

T

pn

D,

wllp

and

Z

W|/t, a

P,"

D’

l=O

inthesettingof

L(a,

,o),where < p < andWis a positiveLebesguemeasurable function definedon(a, *,,). Definitionsand conditions for

P

and

P,

aregiveninthehypothesesofTheorems

1.1 and 1.2,respectively. Wegive conditions on certain

averages

oftheperturbationcoefficients

T-compact. These resultsrely heavilyonTheorems

A

and

B,

whicharespecialcases of Theorem 2.1 in

Brown andHinton [3]. Thesetwotheoremsgivesufficient conditions forweighted interpolation inequalitiesof the form: there exist

dj

>0, r/>0,

K

>0, and e > 0 such thatfor all e (0, e

o)

and y in a class

D

of functions,

where0 <j < n-landl < p < ,,*.

Theorem 1.1 gives integralaverageconditions on

bj

(0 < j < n-1) whicharenecessary

and sufficient for

B

tobeT-bounded or

T-compact

inthecase when < p < and

P

andW satisfy

theconditionsin Theorem 5 in

Kwong

andZettl[6]. When W

=-

1, these conditionsimplythat the coefficients of

T

are bounded above by the corresponding coefficients of an Euler operator.

Furthermore, theperturbationconditions for

T-compactness

of

B

are sufficient for the essential spectrum and Fredholmindex tobe invariantunderperturbationsof

T

by B.

By

definition(Goldberg [4, pp. 162-163]),the essential spectrum of

T,

written

tr,(T),

is the

set of all complex numbers

2

such that the range

R(21

T) of

21

T

isnot closed. The essentialresolventofT,written

p,(T),

isthecomplementof thisset.

By

definition(Goldberg

[4,

p. 102]),theFredholm indexto(T)isgiven by to(T) ct(T)

fl(T),

whereo(T)isthe dimension of the nullspaceof

T

and

fl(T)

isthe dimension

of/.(I)

R(T). or(T) is calledthe kernel index ofT,

and

fl(T)

iscalled the deficiency index ofT.

In

Theorem 1.2, theresults in Theorem 1.1 for thesingle-term operator

T

areextendedtothe multi-termoperator L.

An

nth orderperturbation of

L

is considered in Corollary 1.1. Sufficient conditionsaregivenforinvarianceof the essentialspectrum andFredholm indexof

L

under such

perturbations.

Theorems 1.1 and 1.2 andCorollary 1.1 provide generalizations of results of Balslev and Gamelin[2]aspresentedinGoldberg[4, pp. 166-175]. Theirwork dealswithbounded coefficient and Euleroperatorsin theunweighted settingof

LP(a,

.,,)for < p < oo.

In

Theorem 2.1, thesufficiencyconditions in Theorem 1.1 aregeneralized for operators

T

with arbitrarily large coefficients. Again, these conditions involve integral

averages

of the

perturbation coefficients

bj

(0

<_ j _< n-l). Theorem 2.2 gives pointwise conditions on

bj

(0 _< j _< n-l)under which the conclusions of Theorem 2.1 hold. The case inwhich p is coveredbyTheorem2.2. Also,perturbationconditions which are sufficient forL-boundednessor

L-compactnessof

B

are obtained for the casep and the case in which the coefficientsof

L

are

arbitrarily large. These theorems rely heavily on investigations by Brown and Hinton [3]on sufficient conditions forinterpolation inequalities. Examplesof each theorem are

presented

and contrastedfor the situation in which the coefficient in

T

is anexponentialfunction.

The final theorem, Theorem3.1,deals exclusively with the case p 1. Sufficient,integral averageconditionsaregivenfor T-boundedness ofB.

1.

INTEGRAL AVERAGE CONDITIONS FOR

EULER-LIKE

OPERATORS

In

this section we consider operators whose coefficients arepowers of a fixed functions times a

weightfunctionwand a bounded function.

In

thesimplestcase, i.e., w(t) s(t) 1, Theorem 1.2

donotrequire w(t) or a 0, we refertotheunperturbedoperator of Theorem 1.2 as Euler-like.

THEOREM

1.1. Let < p < and

I

[a, **). Letsandwbe positive,

AC,o

(I)

functions

such that

Is’(/)l

<_

No

and

Is(l)w’(t)l

<-

Mo

w(t) a.e. on

I

for

someconstants

N

and

Mo.

Let ot

R,

W ws

ap,

and

P

ws+"). Let T, B:

l.(a,

**) --->

l.(a,

**) be the maximal operators

corresponding to the

differential

expressions

z

-7

D"

D

and

n-I

Z

bj

D

,

respectively, where each

by

o(l).

For0 <j < n- and

t

> O,

define

D=

j=O

g,.

n(t)

w()s() dT.

Thenthefollowinghold:

(i) BisT-bounded

if

andonly

if

b

L

I)and

sup

g,(t)

< (0 < j < ,-1) (1.1)

aStS*.

(ii)

for

some e O,

l/(2N0)

).

When

(1.1)

holds, the relative bound

for

B

is

O.

Furthermore, the maximal operatorcorrespondingtot

+

v

is

Tr+

T + B.

B

is

T-compact

if

andonly

if

bj

l.

(I)and

lim

g,.(t)

0 (0 <j < n-l) (1.2)

for

some ( O,

l/(2N0)).

When (1.2)holds,

r

and

Tr+

have thesameessential spectrum and

I

e

p,(T)

to(M-T)

r(M-T/o),

where

p,(T)

is the essential resolvent

of

T

andto(T)istheFredholmindex

of

T.

Thefollowingtheorem ispartof Theorem 2.1 in

Brown

and Hinton

[3]. It

givessufficient conditionsforweighted interpolation inequalities.

THEOREMA.

Let <p <.0, l=[a, ..),and 0 < j < n-1. LetiV, W, andPbe positive .nu,zJ.,zd,J,.sio.j.ab .h,

.

:(:),.

j’"

"

"

L

II

-q’,

,’-’’

,,(,5

+

1"

for

p 1, W

-j,

P-

are

locally

essentially boundedon

L Suppose

thereexists

P q

e > 0andapositivecontinuous

function

f

f(t)

on

I

such that

:=

.,,,,,-,,,

for

alle (0,

eo),

where

llP-’il..,,.,/,,.

Lef

Jt

J

p=l

<p<.,,

withsimilar

definitions

for T.(W).

Then thereexists

K

> 0 such that

for

all e (0,

e

o)

and

yeD,

where

D

{

y"

y(n-,,

AC,(1),

I

WlyIP

< *,,, and

I

Ply’’’I

<

"}"

PROOF OF THEOREM

1.1.

(i) Sufficiency.

Suppose

(1.1)holds for some d/ 0,

1

).

We

will show that TheoremA appliestothe choices

f

s, N

Ibj[

’

e

//,

andWand

P

as inTheorem1.1. Basic estimatesare obtainedfrom thefollowinglemma in[3, pp.

575-576].

LEMMAA. Let

sandwbe as inTheorem 1.1. Then

for fixed

I, 0 <

e

< 1/N and

< < t+es(t), we have that

(1-eNo)s(t)

<_ s(r) <_

O+eNo)s(t)

and

exp/-o

)

wCt)< wCz)<

exp(-o

)

w(t).

Thisimpliesthatbothpositiveandnegativepowersof s(z) and w(z) areessentially constant for

< <

+

e s(t)and fixedt.

By

l.emmaAand the definitions of

P

andW,

T.(P)

[:/.,o)

w(.)-qp

s(.)-+,)

d’t"

<- Ct

w(t)-’

s(t)-+’) _(1.3)

and similarly

Tt.

e(W

< C

w(t)-’

s(t) (1.4)

for all e

I

and

e

(0,

d/),

where

C,

and

C2

areindependentof ande. Using

Lemma

A

again,

weobtainfora constantC

ef(t)

e

s(t)

e

s(t) w s

<

C3

w(t) s(t)(a+)

g.(t)

[,+cf(,

N < C w(t) s(t)<=*j)

(1.5) .f(t)

for all e 1,t e (0,

6).

Thus

S(e)

<

sup,,,

{

s(t)<"-J)

C

w(t)-’

s(t)-a+")

C__e

w(t)S(t)("m)p

}

sothat

S,(e)

_<

CC,,

0 <

e

<

6.

(1.6)

Similarly,

S(e)

<

CC:,

0 <

e

<

6.

(1.7)

Hence, byTheoremA,there isaconstant

K

such that for all y e

D

D(T),

Use

oftheelementary inequality

(a

+

bP)

’i < a

+

b (a,b > 0) gives

for all y e D(T), 0 <j < n-l, where

K,

K

’.

Restrict e < 1. Then therightside can be bounded aboveindependentlyofj, and thetriangle inequality gives

for all y e D(T). Since p > 1, itfollows that

B

is T-bounded with relativebound 0. The result

Tr/

T

+ B

followsbyan argument given onpp. 169-170inGoldberg

[4].

Necessity.

Suppose

B

isT-bounded.

Let

bea function in

C’(R)

suchthat on

[0, 1]

andsupport(0) t-2,

21.

Fix

6

0,

l/(2No)

).

For

eachr

>

a,define

t > a. (1.9)

,(t)

6s(r)

Then

Cr

on [r,

r+6s(r)]

and

support(0,)

[r-26s(r),

r+26s(r)].

We proceed

by an

inductionargument. First consider j 0 in(1.1). Fix r > a.

Note

that

B

0r

[r,r

+

t

s(r)], so that

go.(r)

s-r)

WSap

constantC independentofrsuch that

Now,

applying

Lemma

A,there is a

go.

(r)

< C

I

r+s(r) C

w(r)s(r)+ap W

IB

rl

<

w(r)s(r)/ap

lIB

,11’

(11 ,11"

+

lit

rl[

p)

(1.10)

<

w(r) s(r)’+ap

for some constant

K

independentof r, where the lastinequality followsfrom thehypothesisthat

B

is T-bounded.

Using the compact support of

,,

Lemma

A, a change of variable, and the fact that

C0

(R), we have for someconstant C0,

< Cw(r)s(r)a"

Ii

I (u)l"

,(r)du

<

C

w(r) s(r)aE+ (1.11)

for someconstant

CI

independentofr. Similarly, for some C0,

lIT IrlIP

I:

w

IT

rlP

fP

n)lP

Jr-2cs(r) W

dt

<

Cow(r)

s(r)’a+",

C w(r) s(r)(a+")’

p<")(u)

6

_s(r)dt

"s(r)"

< C w(r) s(r)a+’ _(1.12)

for some constant

C2

independent of r.

Use

of (1.11) and (1.12) in

(1.10)

yields

go.,(r)

<

K(C

+

C),

r [a, **). Therefore,(1.1)holdsforj=0andalld; 0,

l/(2N0)).

Nextfix k _< n-1.

Suppose

(1.1)holds for 0 <j _< k- andsome d; 0,

1/(2N

o)

).

k-I

Let

a

be the maximaloperatorwith actiongiven by

a

17

bj

D

By

the sufficiency

1--0

argument above,

A

isT-bounded. Thus since

B

isT-bounded, Minkowski’sinequality impliesthat

A

n-!

+ B

isT-bounded. Notethat (A

+

B)y

W/,

,

bj

yO),

y D(T). With and

,

defined

l=k

h(t)

O(t)

.,

t>a. Then h

C’(R)

and h(k)

-=

on[0,1].

For

eachr > a,define

(1.13)

hr(t

s(r) h(u), > a, (1.14)

hk)(t)

t-r where u

ds(r)"

Then

h:)(t)

h()(u),

support(h

[r-2ds(r),

r

+

26s(r)].

Thus

b

_{on[r, r}

+ 6

_s(r)].

(A

+ B)h,=

W

p

for r < <

r+s(r),

and

(1.15)

By

Lemma

A,weobtain for aconstantC,

g. (r)

r)

ws(+*m

s(r---

W S(Ot+k)p

C

[r+;(r)

w(r) s(r)(a+*m+’

WI(A

+

B)

h,]"

<

C

w(r) s(r)<a/*’’/’

II(a

+

m

h,

ll’

C

<

w(r) s(r)’+’)’+’

llh,’

+

liT

hrll

p

),

(I. 16)

where the lastinequality follows fromthe relative boundedness of

A + B

withrespectto T.

By

calculations like those usedinderiving (1.11)and(1.12),we obtain forr >_ a,

Ilh,

il"

<_

c,

_{w(r) s(r)} (1.17)

and

where

C

and C areconstantsindependentofr. Thus

(1.6)

impliesthat(1.1)holds forj kandany

d

(0,

1/(2N0)).

This establishesnecessityof(1.1).

(ii) Sufficiency.

Suppose

(1.2) holds for some

6

0,

1/(2No)

).

We

will use an argument similartothat inGoldberg

[4, pp. 171-172]. For

eachpositive integer

N >

a, define

B#

{yon[a,N],

on D(T) by

By

We

show that

B

N convergesto

B

in the

space

of

on (N, .o).

boundedoperatorson D(T) withthe

T-norm.

First note that

T

isclosed.

To

see this, let

f,

--

f

and

Tf.

--.

g in

/.,(a,

.,,). Let Jbe acompactsubinterval of[a, *,,) and restrict the functions

f,

f,,

and gto

J.

Define Tj

/.(J)

---)

/.(J)

tobe the maximaloperator correspondingto

z

on J. Clearly,

f.

--4

f

in

/,(J)

and

f.

D(T).

Since

Tf,

(Tf.)l,,

Tf.

g in

L(J).

By

Therefore D(T) is complete under the

T-norm.

D(T) D(B). For y D(T),

From

(i),

B

is T-bounded. So

(1.19)

By

the argument used inprovingsufficiency in(i),Theorem

A

appliestothe interval I [N, .o)

withthesame choices for the weights,

f,

and

e

0.

By (1.3)

and(1.4),for 0 < e <

S(e)

< C sup

{w(t)-’s(t)

-/)

[,/.,O)]bl,

}

Iv.**)

e

s(t)

(1.20)

and the same estimate holds for

S(e)

up to a multiplicative constant.

By

Lemma

A, for

O<e<d/,

[,+.,,,,

-<

c

w(t)

s(t),a/,,,

g,.n(t)

IN,

oo). (1.21)

es(t) e

Hence

C

S(e)

<_ _sup

g,.n(t)

_(1.22)

E

with asimilar estimate for

S:(e),

0 <

e

<

6,

where

C

isa constantindependentof

N

and e. It

follows from Theorem

A

thatfor ally e D(T),

I’

b,

Y"’I"

<--

I

W

IY[

p

+

I?

P

’(’)

where

Cj

isindependentofyandN (but dependsone).

Use

of(1.23)in

(1.19)

gives

IIy-

>1

_{_<}

Z

c,

sup

gj.,(t)

Ily]l

,=o ,,,.-,

(1.24)

for all y D(T)such that y 0.

By

(1.2),thetermontherightsideapproaches0 as

N

---> ,,*.

Therefore,

Bu

--->

B

in thespaceofbounded operators onD(T)withtheT-norm.

Next,

we show that each

Bu

is

T-compact.

Let

{ft}

be a T-bounded

sequence,

say

IIf,

L

-<

’

forall I. Wewillshow that

{fJ’},

0 <j < n-1, is uniformly bounded on [a, N].

Partition

I

[a,N]

by

J,

[t,,t,+],

< < k, with

t

a,

t,+

t, +

es(t,),

and

e e (0,

d)

chosen such that

N

tk+

tk +

es(t).

From

theproof ofTheorem 2.1 in

Brown

and Hinton[3],with

J,,

f

If,,,,<,)l

_<

[s<,,>]-"

z

(w>

I,,

wlf,

l"

+

[es(t,)]

’"-’)p

T

,(P)

s(t,)

I,,

Use

of(1.3)and(1.4) yieldsfor someC (dependingone),

w(t,) s(t,)

{I,,

wlzr

+

I,,

for E

J,.

Sincewand s arepositive,continuous functions on [a, ) and t,

J,

c: [a, N], we havefor someC dependingone,

(1.25)

for [a,N], 0 <j < n-l. Since

{f}isT-bounded,

{f(’)},

0 <j < n-l, isuniformly bounded on[a,N].

Next

we show

{f(s)},

0 <j < n-1, isequicontinuouson [a, N].

Let

r/ > 0 be given.

For

t,s e [a, N],

If,<"(:

t,

-<

LFI

IS:

w

’’"

byHolder’s inequality, where

+

1. Sincew andsarepositive, continuous functions on

P q

[a, oo),

W

ws

"’

isboundedabove andbelowon [a,

N]. Hence

fort, s e [a, N],

If,<’>(t) -f,<"(:’)t

<

c

lt-"

(1.26)

where theconstantC dependsonW. Forthe case 0 <j _< n 2, theargumentused toobtain

(1.25) appliesto j

+

< n andyields

If,/"(t

_<

c

IV,

L,

[a, N]. Thisimpliesthat, sinceWisboundedon[a,N],witha newC,

ils,<,+’>ll

<

c

ILf,

L.,

o

<j < n- 2 (1.27)

L;(a.N)

Forthecase j n 1,

(")

Ils,<’+"ll,,.<o.,,

Ils,

L.<o.,,

<-

wiry,

<

c

rs,

<

c

IIs,

L

(1.28)

since W/P is bounded on[a, N]. Thus,inanycase,(1.28)holds for 0 <j < n 1.

So (1.26)

impliesthat

If/"(’) -f,("(s)!

-<

c lt-

,1"

IIs;,ll,,

<-

M

it-

1",

(1.29)

where

M

c

s.p{ IiZL

1>

1},

since

{fl}

is T-bounded. Since p > 1, llq >

O.

Therefore,

{f(J)}

isequicontinuousandboundedon[a, N], 0 < j < n 1.

By

theArzela-Ascoli

Theorem,

{f}

has a subsequence

{f,

0}

whichconvergesuniformly on [a, N],and

{f0}

has a

uniformly on [a, N]. Repeatingthisprocedure,asubsequence

{g,}

of

{ft}

is obtained such that for 0 <j < n 1,

{gJ)}

convergesuniformly on[a, Nq.

By

definitionof

B,,

(1.30)

It

followsthat

{B

g,}

converges in/_.(a, .0)

as

--->

**. Thus

Bn

is

T-compact

foreachN,and so

B

is

T-compact,

beingthe uniform limitof

T-compact

operators.

Necessity.

Suppose B

is

T-compact.

First we showthat(1.2)holds for j 0. We proceed byacontradiction argument.

Suppose

that forany d; 0,

l/(2N0)

),

there exists e > 0 and a

sequence

{r}7__i

ofpositivenumbers such that r--> and

> e, > 1. _(1.31)

Fix

t

O,

1/(2No)

).

Let

{,}

be the functions definedby (1.9). Asbefore,

Bi,

W,,

b0, on[r, r

+

6s(r)].

(1.32)

Itfollows from(1.31)andLemma

A

that

[

I’

e <_

r,+,<r,

W

l,,

<

s(r)

wS

w(r,) s(r,)

’+a"

whereC is aconstantindependentof

I.

Foreach r > a, define

I[Ir(t)

w’r’l/’t

s

,(t),

_

[a, 0.). (1.34)

Then

and(1.33)implies that

subsequence. Relabel indices so that

{Br

}

converges

in

L(a,

**)to some

Yo. We

show that

Yo

0 a.e. in [a, *,,). Let

Jo

be a finite subinterval of

In,

-0). Since

r

---> as --)*,, and

support(Vr,)

[r

2

ts(rt),

r

+

2

ts(r)],

wehave

I//r,

m 0 on

Jo

and

Br

0 on

Jo

for

<

IlYo

BI//,,Ii.

Since

Brr,

-’>

Yo

as --> and thetermon the left sideisindependentofl,

Ilyoll ,,o,

o.

This holds for anarbitraryfinite subinterval

Jo

of

In,

-0),andso

Yo

0 a.e. in

In,

-0). Therefore,

Blp’,,

-->

0

in

L(a,

*,,)as -->

.,,.

This contradicts

(1.36).

Thus

(1.2)

holdsforj 0.

To

establish (1.2) for _<j_< n- 1, we use an induction argument. Fix k _< n- 1.

Suppose

(1.2)holdsfor 0 _< j _< k and some

t

0,

1/(2No)

).

Suppose

(1.2)doesnothold for j k. Then there exists e > 0 and a

sequence

{rt}

ofpositivenumbers such that

r

--

as

g,.(r)

>_ eo, >_ 1. (1.37)

As in the proof of necessity in (i), let A be the maximal operator with action defined by

k-i

A

,,

X

bj

D

.

Then

A

is

T-compact

bythesufficiencyargument in(ii). Since

B

is

T-compact,

B

isT-bounded. Therefore, the estimate preceding

(1.16)

yields, with

h,

as in(1.14),

w(r)

s(r)

"+)/’

For

each r > a, define

pr(t)

w(r),/,

s(r)++j/"

h,(t),

> a. (1.39)

Then

g. ,(rt)

<

C

(A

+

B),r,

ll’

(1.40)

and

I,,,

w<r,

,<r,

:"/"’

(liar,

+

limb’,

II)"

.41)

By

(1.17)and(1.18),

{p,,

}

isT-bounded. Since

A

and

B

areboth

T-compact,

A + B

is

T-compact.

Therefore,

{(A

+

B)p,,}

contains a convergent subsequence,

say

(after relabeling indices) (A

+

B)p,,

-->

zo

in/.(a,

**). Weshow that

zo

0 a.e. on[a, **).

Let

Jo

c [a, **)bea finite interval. Since

support(p,)

[r 2

d;s(r),

r

+

2

d;s(r)],

p,,

=-

0 on

Jo

and hence

(A

+

B)p,,

0 on

J0

for all sufficiently large.

For

suchl,

Thus

j,,

W

Izo]

0 foranyfinite subinterval

Jo

of[a, ,). Therefore,

Zo

0 a.e. on [a, oo)and

(A

+

B)p,,

0 in

L(a,

.,,).

Hence

(1.40) implies that

gk.(r)

---> 0 as

contradicting (1.37). Therefore,(1.2) holds for k. This establishes necessity of(1.2) for

T-compactness ofB. Thus Theorem 1.1isproved. 1

Note

that Theorem 1.1 deals withperturbations of asingle-term operator T.

In

thenext theorem, we extend Theorem 1.1toa multi-termoperator L.

THEOREM

1.2. Let p, s, w, W, P, B, and _gj.s be as in Theorem 1.1.

L"

Ifw(a,

**) -->

Ifw(a,

**)be the maximal operator correspondingto

Let

lip

W,p

a,

P,

D’,

wherem, _ai (0 < < n) L"(a, **)and

P,

wst/p. Then the following hold:

an

(i)

B

isL-bounded

if

andonly

if

bj

o(a,

**)and

sup

gj.s(t)

< (0 <j < n 1) (1.42)

at<*

for

some

d

(0,

l/(2N0)).

When (1.42) holds, the relative bound

for

B

is O.

Furthermore,the maximal operatorcorrespondingto

+

vis

/v

L +

B.

(ii)

B

is

L-compact

if

andonly

if bj

o(a,

.0)and

lim

gj.s(t)

0 (0 < j < n- 1) (1.43)

for

some

(0,

l/(2No)).

When

(1.43)

holds,

L

and

Lt/

have the same essential spectrum and /%

p,(L)

r(M-

L)

r(M-

Lt/,).

To

prove

Theorem 1.2, we will use thefollowinglemmas.

LEMMA

1.1. SupposeA, C, andDarelinearoperators such thatDis C-boundedwith relative bound less than 1.

(i)

If

A

isC-bounded, then

A

is(C

+

D)-bounded. Furthermore,

if

the relativebound

of

A

with respecttoCisO,then the relative bound

of

Awith respecttoC +

D

isO.

(ii)

If

Ais

C-compact,

thenAis (C

+

D)-compact.

PROOF.

For (i), we have D(C)

_

D(D), D(C) c: D(A),

IlDyll-<

K,

IlYtl

/ e

llcMI

(y D(C)) forsome

K

>

0and

e

(0, 1), and

IIal

-<

K=

IlYtl

/

,llcti

(y D(C)) for some

K=,

d

> 0. Therefore, D(C

+

D) D(C) c:_ D(A). Fix y D(C). Then

IIAyll-<

K

INI

+

all(c

+

O)y-

o>tl-<

c

Ilyll

+

&lKc

+

o)ytl

+

&lloyll

Noting that

Ilcyll-<

IIc

+

D)I

+

IIZl

-<

IC

+

oYll

+

K,

IlYll

+

ecl,

,

obtain

Ilcyll-<

II(c

+

D)

+

IlYl.

Hence

IIAyll

K

IlYll

+

I[(c

+

D)MI,

where

1- 1-

1-K

is

indendent

of y. Therefore,A is(C + D)-boundedand thestatementconcerningrelative bounds followseasily.

For (ii),

suppose

{Yn}

is (C

+ D)-bounded,

i.e.,

y,

D(C

+

D) and

]lYnll

+

[[(C

+

D)yn[

<_

K

for some constant

K

independent of n. Then

Yn

D(C) and

Ilcyll

<-II<c

+ D)y, +

Dynll

<

K +

K,

Ily,

+

e]Cy,

by the C-boundednessof

D.

Since 0 < e < and

ilY,

II-<

K

w ha

IICy,

<

K

(1

+

K,)

Therefore,

{Yn}

is C-bounded.

1-e

Since

a

is

C-compact,

{ayn}

containsaconvergent subsequence. Since

{Yn}

was an arbitrary (C

+

D)-bounded sequence,

A

is (C+ D)-compact. 1

LEM1VIA1.2. LetB,

L

and

T

be the operatorsinTheorems1.1and1.2. Then:

(i)

B

isL-bounded

if

andonly

if

B

isT-bounded. Further, the relative bound

for

B

with respect to

L

is0

if

andonly

if

the relativebound

for

B

with respectto

T

is

O.

(ii)

B

is

L-compact

if

andonly

if

B

is

T-compact.

PROOF.

Consider the differentialexpression l-’r

W

Ip

p,

tl,

D’.

Its

I--0

coefficientssatisfytheperturbationconditions(1.1)sincefor e

I

and0 < < n 1,

p, _<

(constant). s(t) _la, w s"+’) s(t)

--,

a,

(0

< < n-

1)

e

L"(1). Hence

byTheorem1.1(i),

L- T

an

is T-boundedwithrelative bound 0. Application of

L

emma 1.1 (with

A

D

/-_}L-

T

and C T) yields that

L-

Tis

T +

L- T

L-bounded with relative

bound 0.

(i)

Suppose

B isL-bounded. Then

B

is

(-,/L-bounded

since L"(1). Another

an

applicationof

Lemma

1.1 (withA =B,

C

ml

L, and

D

T-

__1

L) shows thatB

isT-bounded.

Next, suppose BisT-bounded.

By

Lemma 1.1 (with A B, C T, and

D

is The statementaboutzerorelative

bounds also follows from

Lemma

1.1.

(ii) Thispartisprovedin asimilar mannerusing

Lemma

1.1(ii).

I

(i) Sufficiency.

Suppose

(1.42)holdsfor 0 _< j _< n landsomet$ 0,

1/(2N0)).

By

Theorem 1.1(i),

B

is T-bounded with relative bound 0.

Hence Lemma

1.3 impliesthat

B

is

L-bounded with relative bound 0. The result

D(//v)

D(L) followsbythe sameargumentused in

showingthat

D(T/v)

D(T)intheproofof Theorem 1.1.

Necessity.

Suppose

B isL-bounded. Then

B

is T-bounded by Lemma 1.2. Henceby Theorem1.1,

bj

(0 _< j _< n 1) satisfy(1.42)forsome t$ 0,

1/(2N0)).

(ii) Sufficiency. S.lppose(1.43)holdsfor 0 j P- andsome

/0,

No

)

Then

byTheorem1.1,

B

is

T-compact

andhence

L-compact

by

Lemma

1.2. The invarianceofthe essential spectrum and FredholmindexofLunderperturbations by Bfollow as in theproofof Theorem 1.1.

Necessity.

Suppose

B

is

L-compact.

Then

B

is

T-compact

by

Lemma

1.2.

By

Theorem 1.1, there exists t$ 0,

1/(2N0)

suchthat

bj

(0 _< j _< n 1) satisfy(l.43). 1

REMARK.

Theorems 1.1 and 1.2applytooperators

T

and

L

with coefficientseventuallybounded abovebythecorrespondingcoefficientsofanEuleroperator. Tosee this,notethat thehypothesis

s’(t)l

-<

No

a.e. on

I

impliesthat there exists apositiveconstant

C

such that s(t) < C for all

sufficiently large. Now, by definition of

P,

and W and the hypothesis that

a,

(0 < < n)

L"(1),

we have

la,(t)l

,(t)

s(t)’

<

C,

t’

(1.44)

for all sufficientlylarge,where

C,

areconstantsindependentof and 0 < < n.

EXAMPLE

1.1. Let n 2, p 2, w 1, tz 0, andsbeanypositive,

AC,o([a,**))

function such that

Is’(t)]

_<

No

for

I

[a, **). Then

W

and

P,(t)

s(t)2’

for i=0, 1, 2. Consider

Ly

a2(t)

s(t)

y"

+

a,(t)

s(t)

y"

+

ao(t

y (1.45)

and

By

b(t)

y"

+

bo(t)

y, (1.46)

where --, ao,

a,

a L** (I)and b0,

b

/.

(I). Then

a2

f’/’"’"

Ib(’r

_d’r

gj.

(t)

s-s(’r)2j (j=O, ). (1.47)

By

Theorem 1.2,

B

isL-bounded if andonlyif sup

g.

(t)

< (j 0, and

L-compact

if and

tel

Nextwe prove acorollaryof Theorem 1.2 in which an nth order perturbation

B

ofLis considered. The perturbation is such that the coefficients of thehighest-ordertermsin

L

and

L

+

B

obeythesamehypotheses. Before statingthecorollary,weprovealemmaconcerningthedomains of the single-term operator

T

and multi-term operator L.

LEMMA

1.3. LetTand

L

beas inTheorems 1.1and l.2. Then D(L) D(T).

PROOF.

First consider the case in which a

-=

1.

By

Theorem 1.1 with

v

Wtp

a,P,PD’,B

is T-bounded and

L

T/

T+B. Thus D(T) D(B),and so

D(L)

D(T + B)

D(T).

For

general a such that an, 1/a

L’(1),

we may replace

T

by

a, T

withoutaffecting T-boundednessofB.

It

followsthat D(L) D(a T) D(T).

I

COROLLARY

1.1.

Let

p,s, w,W,

P,,

and

L

be asinTheorem1.2.

Let B:

lYw(a,

**)

-->

IYw(a,

0.)

be the maximal operatorcorrespondingto

WIIp

where bn,

L’(I),

bj

I.(I)

(0

<j < n),

an

+

b

(1.48)

and

:rn.

s(t--S

w(r)sfr)’a+’)’ dr 0 (0 < j < n-l) (1.49)

for

some

(0, l/(2N0)).

Let

R"

l.(a,

**) -->

l.(a,

**) be the maximal operator

corresponding to

+

v. Then D(L) D(R),

r,(L)

eye(R),

and

/%

pc(L)

r(M-

L)

r(M-

R).

PROOF.

In

view ofTheorem 1.2, it suffices to

prove

the corollary for the operator

R

L +

b.

P2/’

D

n.

As

inTheorem 1.1, let

T"

L(a,

,,*) --->

L(a,

.,,)be the maximal

lip

operator corresponding to

r

P

D

n.

Then

R

L +

b T.

By

Lemma 1.3,

D(L) D(T) and D(R) D(T).

Hence

D(L) D(R).

For

any scalar

A

and

y D(R) D(L),

(M-

R)y

.y-

Ly-

b

nP’.‘"

/%y_Ly+

b

{A.y_Ly+

n-,

PERTURBATIONS OF nTH ORDER DIFFERENTIAL OPERATORS 63

where

At

and

S;t

are the maximal operators associated with

(1

+

a/(AJ-

and

-

a

,

D

A

WI/p

bn

p

b I, respectively.

An

applicationof Theorem 1.2(with L, B, and

an

an

L/

replaced by

A;t,

St,

and

AJ-

R, respectively) yields that

Sx

is

Aa-compact,

cr,(Ax)

O’,(M-

R),

and

0

p,(Aa)

=

K’(Ax)

g’(:l/-

R).

(1.50)

By

definition of

Ax,

RI_L=

(

a,

I

a,

an

+

bn

A.

Let

h

an +bn

Then h,

1/h

L’(1)

and

R(R/- L)

{hg"

g

R(A)}.

Theresultthat

R(A)

isclosedifandonlyif

R(M

L)

isclosedfollowsfrom thenextlemma.

LEMMA

1.4. Let

M

beaclosedsubspace

of

lfw(a,

**)and

N

hM

{hg"

g

M},

where h, 11 h

L"(a,

oo). ThenNisclosed.

PROOF.

Suppose

hg

N

with

gn

e

M

and hg

--

z.

Since 1/h

L"(a,

00),

’gn

-’> z/h. Since Misclosed, z/h M. Therefore,

z

h.(z/h)

N.

So

Nisclosed. 1

Since

tr,(Ax)

o’,()I/-

R), p,(Ax)

p,(itl-

R),

i.e.,

{/z"

R(/Z/

A)is

closed}

{/2"

R(/z/

(R/

R))is

closed}.

Therefore,

R(A:t

closed =:,

R(,qJ

R)

closed. Itfollows that

p,(L)

p,(R);

and so

o’,

(L)=

It remains to show that it e

p,(L)

=#

(AJ-

L)

()d- R).

Let

it e

p,(L).

Then

R(A/- L)is

closed and

L(a,

*,,)

R(it/- L)

M, where

M

N(I- I.:).

Since

L*y

ity hasat most n

L(a,

oo)solutions,

M

isfinite-dimensional.

an+b

Let gt= Then

,

L"(1)

and

A

(;t/-

L).

Any

f

/.(a,*,,)

can

a

be written as

f

(:hr

L)g

+

m, whereg e D(L)andm M. Thus

with

vf

l.(a,

**),

(/1I-

L)g

R(A),

and

gan

vM.

Now,

since

R(ft/-

L)

closed

(a)

losd,

<a,

**)

(a)

*

N

wh S--

Vt

{"

m

L’*(a, *,,), dimN dimM.

By

definition, the deficiencyindexofA is

fl(Ax)

dim[/.(a,-0)

R(Ax)

dimN dim

M

dim[/.(a,

oo)

R(t/-

L)]

fl(:t/-

L).

Since

A;t

gt(M

L)

and Ig,: 0 (because L’(a, **)),

N(Ax)

N(Zt-

L).

Therefore,

ct(Aa)

ct(M

L).

Thus

tc(At

K’(M- L).

Since

R(Ax)is

closed, 0

p,(ax).

Hence

by

REMARK.

Notethat(1.49) and(1.43)are identical conditions on the lower-order perturbation coefficients

bj,

0 <j < n 1. Theorem 1.2 is a result for lower-order perturbations of

L

Wi/p

a,P,

D’,

whereto,

a,

(0 < < n 1) E

L’(a,**).

Corollaryl.lappliesto

nthorderperturbationsofLoftheform

R

a

+

bn)

P"

D +

2

(a,

P,l’"

+

b,)O’

I=0

where b satisfies(1.48)and

a,

+

b,,

L’(a, **) (inanalogytothe conditionson

a

intheoperator L).

2.

CONDITIONS FOR OPERATORS WITH LARGE COEFFICIENTS

Recall that Theorem1.1appliestooperators

such that

P(t)

]’

_{< Ct} W(t)J

forsomeconstantCandall sufficientlylarge. Thefollowingtheorem generalizes thesufficiency

conditions inTheorem 1.1for operators

T

witharbitrarilylargecoefficients.

THEOREM

2.1. Let < p < and I [a, **). Let

P

and

W

be nondecreasing, positive

continuous

functions

on

I

such that W

-qlp,

p-q/i,

L(1),

where

--+--

1. Let T,

P q

B

L

(I) --> Ifw(I) be the maximal operatorscorrespondingto

pIIp

7

-

D

and

n-I

j=O

respectively, where each

b

E

o(l).

For

0 <j < n and

>

O,

define

(0

If

thereexists

5 >

0 such that

sup/z.(t)

< (0 <j < n- 1), (2.1)

tel

(ii)

If

thereexists > 0such that

lim

/j.(t)

0 (0 <j < n 1), (2.2)

thenBis

T-compact.

PROOF.

(i)

Suppose

(2.1)holds for some > 0. Wewillshow that TheoremA appliesto

the choices

f=

N

’,

and e

.

Fix

I

and e

(0,).

SincePis

nondecreasing onI,itfollows that

/,)

dr <

T,,,(P)

ef(t)

P(’)/"

P(t)

Similarly,

T,

(W) < The choice

f

ismade so thatcertainupperbounds on

w(t)

S,(e)

and

S2(e)

are equal:

Sk(e)

<

,

sup/zj.(t)

(k 1, 2).

By

(2.1), there exists a

tl

,constant C independentof such that

Sk()

< for k 1, 2 and (0,

).

Hence by

TheoremA,there is aconstant

K

such that

for all y D(T).

By

the same calculations usedto obtain (1.8)in the proofof Theorem 1.1,

IIyll-<

K,

’-+’-""

IlYll

/

K,

:-""

IITMI,

K,

K

’’p,

for all y D(T). Since p > 1, the coefficientof

IITMI

canbe madearbitrarily smallby choosing

e

e (0,

)

sufficiently small. Therefore,BisT-boundedwithrelativebound0.

(ii)

Suppose

(2.2)holds for some

d

> 0.

T-compactness

of

B

followsbytheargumentused in proving sufficiency in Theorem 1.1 (ii). 1

-I

EXAMPLE

2.1. Let W(t)

=-

and P(t)

e’.

Then T e’/’D" and B

bD

.

In

this

j--0

case, condition

(2.1)precludes

exponential growth of

bj.

Suppose

Ibm(t)

<

Cjt

a’,

a < < 00, 0 < j < n-I,

A

> O,

for some constants

C

and

A.

Fix j and let

A

A

and C

C:.

Thenbythe definition

of/z.

in Theorem 2.1,

e(S/,,

/(,,,))

[.:

z

dz

(Ap

+

I)e0/" ’/"))’

+

de’/’))

For

sufficiently large,we obtain(witha differentconstant)

e(p i)tl(np)

Ce(a-J)t/n

n-I

Hence

(2.1)holds if

A

_<j,and

(2.2)

holds ifA < j.

For

example,the Euler operator

D’

is

1--0

T-bounded, and the operator

[

j-D

(e >

0)

is

T-compact.

I

j=0

Westatehere another part of Theorem 2.1 fromBrownand Hinton[3]mentionedearlier.

THEOREMB.

Let _< p < ,,o,

I

[a,,), and 0 <_ j <_ .-1. Let N, W, and

P

be positive measurable

functions

such that N e

Lt.(1);

for

p > 1, W

-"1,

P-q

I.(1)

where

+

1;for

p 1, W

-,

P-|_{arelocallyessentially boundedonI.}

Define

P q

T,.

(t’)

p=l

<p<,,

withsimilar

definitions

for

T.

(W).

Suppose

thereexistse > 0 andapositivecontinuous

function

f

f(t)onIsuch that

f’(t)

> O,

R,(e)

:=

sup{f(,)

("-)_N(t)

T.t(P)}

_<

tl

and

R2(e)

:=

sup{f(t)

-’

N(t)

T,.

,(W)}

<

t!

for

alle (0,

e0).

Then thereexists

K >

0 such that

for

alle (0,

Co)and

y D,

I,

rCly’ ’l

<-

r

I,

W[Y]

+

et’-"P

R,(e)It

P

where

D

{y"

y"-"

ACto(’),

t

WlY

<**, and

I,

<**}

This result canbe usedtoprovethefollowing theorem,whichgivea pointwiseconditions sufficientfor relative boundedness and relative compactness.

THEOREM

2.2.

Suppose

the conditionsin Theorem2.1are

satisfied

with the

definition of lz.

replaced by

In

addition, suppose

w

P

AC(I)

with d

L-j

P(t)

>-

0

for

I. Then the conclusions in

Theorem 2.1hold

for

<_ p < providedthat

for

the case p 1,

W

-

and P-| _{are locally}

essentiallybouledonI.

PROOF.

(i)

Suppose

sup

uj(t)

<

for 0 _< j _< n-1.

We

willshowthatTheorem

B

P and W are nondecreasing on I,

T,.(P)

< and

T,.(W)

<

Hence

P(t) W(t)

R,(E)

<_

Tpt

f(t)

("-’)"

[0,(t)

l’

e--SS

and

P(t;)

_<

sup

f(t)-’"

[0,(t)["

By

the choice

off, R(e)

_< sup

pj(t)

< (k I, 2). Therefore, Theorem

B

applies. Therestof theproof,

tel

including (ii),follows as in theproofof Theorem2.1.

I

EXAMPLE

2.2. Let W(t) and P(t) e ct > 0. Then

T

e=’/’D

and

B

bj/Y.

Let < p <

,.

Suppose

lb,</)l

_<

c,

ca,’

a < < 0 < j < n-1 for

j=O

Ib,</)l

<

c;

e(#j’-’’")’ ThusbyTheorem 2.2, someconstants

Cj

and

fls"

Then

/s(t)

_,,/,

e

fl

<

otj

BisT-bounded and

fl

<

ctj

=>

BisT-compact.

np np

So the pointwiseconditionson b in Theorem 2.2 allow

b

to grow exponentially.

In

contrast, theintegral

average

conditionsof_{Theorem 2.1 appliedto}

Example

_{2.1 allow polynomial,} butnot exponential,growth of

b.

1

,3.

INTEGRAL AVERAGE

CONDITIONS

FOR

THE

CASEp

₁

Thefollowingtheoremgivessufficient conditions forT-boundednessfor the case p for integral averages.

THEOREM

3.1. Let

P

andWbenondecreasing,positivecontinuous

functions

such that1and

1__

P

W

arelocallyessentially boundedon [a, **). Let T, B:

l(a,

**) -->

lJw(a,

**) be the

maximal.

operatorscorrespondingto

and

"c

PD"

W

n-I

V=

W

bDJ

respectively, where each

b

is ameasurable

function

on [a, ,). For0 <j < n- and > O,

define

r,,,,,,,:,:,l<,+,,,:

,,+.,r,.<,,1

,’,..":’:>:

j,

,-’<,>-,

If

thereexists

>

0 such that

sup

/z,.(t)

< (0 <j < n-l),

thenBisT-bounded.

If

in addition

b_

=-

O, then therelativebound

of

Bwithrespectto T is O.

PROOF.

Weshow that Theorem

A

appliestothe choices

f

p 1,

N

andany

e

t.

Fix e [a, .,,) and

e

(0,

t).

Usingthe

hypothesis

that

P

is nondecming,

Similly, (W)< These inequalities

we have

.,(P)

-.It.,+ef(,)l P(t) W(t)

yield

upper

bounds for

S(t)

d

S(t).

The choice

f

is made so that these

upr

bounds eequal: fork or 2,

S(e)

sup

g.n(t)

<

M

where the last inequality

followsbyhypothesis(forsomeconstant

M > 0). By

TheoremA,there exists

K

> 0 such that for all e (0,

iS)

and y e D(T),

Let denote the norm

of/2w(a,

,,,,). Then

j=O j=O

n-1 n-!

< g

y.

{

-

S=<>

IM

/

-

s,<>

IITM!

}

< g g

{

--’

IlYil

/

--’

IITMI

}

j=O 1=0

where wehave used the estimates on

St

and

S=.

Hence B

isT-bounded.

If

bn_

0, then the previous sum can be truncated at j n- 2:

KMf

e*"/IITy{I

for all

y

D(T), where C(e)is independentof

C(e) y.

\j--0

Restrict e (0,

d)

such that e < I. Then

IIB3I

<

C(e)I13I

+ K M

(n

I)ellT3l

for all y D(T), fromwhich itfollows that the relative bound of

B

with respecttoTis 0.

ACKNOWLEDGEMENTS.

Supportedinpart bythe Universityof

Tennessee

Knoxville and Oak

RidgeNationalLaboratoryScience Alliance

Program.

The author would like to thankProfessor Don B.Hinton for his advice.

REFERENCES

T.G. Anderson.

A

Theory

of

RelativeBoundednessand Relative

Compactness

for

Ordinary

Differential

Operators. Ph.D..thesis, Universityof

Tennessee

Knoxville

(1989).

E.

Balslev and

T.W.

Gamelin. The Essential

Spectrum

of aClassofOrdinaryDifferential

Operators.

Pacific

J.

Math.,14

(1964),

755-776.

R.C.

Brown

and

D.B.

Hinton. Sufficient Conditions forWeighted InequalitiesofSum

Form.

J.

Math. Anal.Appl., 112 (1985), 563-578.

S.

Goldberg.

UnboundedLinear

Operators:

TheoryandApplications. (NewYork:

Dover,

1985).

T.

Kato.

PerturbationTheory

for

Linear

Operators.

(Berlin:

Springer-Vedag,

1966).

M.K.

Kwong

andA.Zettl. Weighted

Norm

InequalitiesofSum

Form

InvolvingDerivatives.

Proc. Roy. Soc.

Edinburgh Ser. A., 88 1981 ), 121-134.